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Vandermonde sets, hyperovals and Niho bent functions

  • * Corresponding author: Kanat Abdukhalikov

    * Corresponding author: Kanat Abdukhalikov 
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  • We consider relationships between Vandermonde sets and hyperovals. Hyperovals are Vandermonde sets, but in general, Vandermonde sets are not hyperovals. We give necessary and sufficient conditions for a Vandermonde set to be a hyperoval in terms of power sums. Therefore, we provide purely algebraic criteria for the existence of hyperovals. Furthermore, we give necessary and sufficient conditions for the existence of hyperovals in terms of $ g $-functions, which can be considered as an analog of Glynn's Theorem for $ o $-polynomials. We also get some important applications to Niho bent functions.

    Mathematics Subject Classification: Primary: 51E21, 94A60; Secondary: 51E15, 51E23.

    Citation:

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  • Table 1.  Set $ \mathcal{D} $ in small fields

    $ q $ Elements in $ \mathcal{D} $ $ \mid \mathcal{D} \mid $
    4 1 1
    8 1, 3, 5 3
    16 1, 3, 5, 7, 9, 11, 13, 37 8
    32 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 69, 73, 77, 85, 89,147 21
    64 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61,133,137,141,145,149,153,157,165,169,173,177,181,185,275,281,283,291,297,299,307,313,409,425,661 55
    128 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99,101,103,105,107,109,111,113,115,117,119,121,123,125,261,265,269,273,277,281,285,289,293,297,301,305,309,313,317,325,329,333,337,341,345,349,353,357,361,365,369,373,377,529,531,537,539,547,553,555,561,563,569,571,579,585,587,593,595,601,603,611,617,619,625,627,633,785,793,809,817,825,841,849,857,873,881, 1093, 1095, 1107, 1109, 1111, 1123, 1125, 1127, 1139, 1141, 1301, 1317, 1333, 1365, 1381, 1587, 1619, 2341, 2349, 2381, 2405 147
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  • [1] K. Abdukhalikov, Bent functions and line ovals, Finite Fields Appl., 47 (2017), 94-124.  doi: 10.1016/j.ffa.2017.06.002.
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